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And no more than that.
If pi contained a subset of digits identical to its own decimal expansion, then the sequence of digits before the subset would be repeated within the subset; and the subset would contain a similar sequence of digits repeated within a further subset ... And so on. Which would mean that pi was a repeating decimal, which would mean it could be expressed as a ratio, which would mean it was a rational number. Which it isn't. Therefore pi cannot contain a subset of digits that repeat its own decimal expansion.

Good point. If you're willing to just encounter the digits, skipping over the digits that don't match rather than regularly skipping a fixed-size gap, then you will be able to re-find the digits of pi, just not consecutively.

Containment can have different levels. My own fetish number (down there in my signature, the Thue-Morse sequence), an infinite non-repeating sequence, does contain itself in a way. Regard the pattern of "01" and "10" pairs you see, or the "0110" and "1001" quads. Both (and infinitely more) appear in the same pattern. If you change every 01 to zero and every 10 to one, you get the same infinite sequence.

Oh, I neglected to mention the more obvious containment of the Thue-Morse (see signature) sequence within itself. It is exactly as Chuck wondered about pi. From Thue-Morse, take every other digit. You get Thue-Morse. Take every 4th digit. Take every 8th, or 16th. You get Thue-Morse again. Every 32nd, etc.

Again, this is a nonrepeating infinite sequence. And it's so nonrepeating that it completely avoids any triples, any repetition of one group three times: no 000, no 010101, no 101101101, etc. Way cooler than a number deserves to be.

And no more than that.
If pi contained a subset of digits identical to its own decimal expansion, then the sequence of digits before the subset would be repeated within the subset; and the subset would contain a similar sequence of digits repeated within a further subset ... And so on. Which would mean that pi was a repeating decimal, which would mean it could be expressed as a ratio, which would mean it was a rational number. Which it isn't. Therefore pi cannot contain a subset of digits that repeat its own decimal expansion.

Grant Hutchison

Infinities do strange things. The infinitely long expansion of pi could contain an infinitely long expansion of pi...

I'll go take my meds now.

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Infinities do strange things. The infinitely long expansion of pi could contain an infinitely long expansion of pi...

Well, that's what I was talking about - an infinite subset, which in turn must contain its own identical infinite subset, which in turn contains another identical infinite subset, and so on. That would make it a repeating decimal. But pi is an irrational number, which means its non-repeating. Therefore it can't contain such nested identical infinite subsets.

If Pi had been exactly equal to 3, would the history of the development of mathematics out significantly different in any way from OTL?

I'm quite aware of that; it can't be represented by a finite expansion. Infinities behave strangely: if I number each one of the real numbers between 0 and 1 with a whole number, there would be a one-to-one correlation between 0<x<1 and all whole numbers. This can be repeated between any two arbitrary real numbers that aren't equal, e.g., 0.5<x<0.6.

Information about American English usage here and here. Floating point issues? Please read this before posting.

I'm quite aware of that; it can't be represented by a finite expansion. Infinities behave strangely: if I number each one of the real numbers between 0 and 1 with a whole number ...

Stranger than that, even.
The set of real numbers between 0 and 1 is uncountably infinite, so you can't put it into a one-to-one relationship with the set of whole numbers, which is countably infinite.

I love irrational transcendental numbers too. Somewhere in pi there are a billion zeroes in a row.

And if you assign each triplet of digits as the ASCII codes for characters, pi somewhere contains this post, your post, in fact the entire forum, plus the DNA of everything living creature on the Earth, plus our entire history and every book ever written and every conversation ever engaged in. An infinite number of times.

And if you assign each triplet of digits as the ASCII codes for characters, pi somewhere contains this post, your post, in fact the entire forum, plus the DNA of everything living creature on the Earth, plus our entire history and every book ever written and every conversation ever engaged in. An infinite number of times.

And if you assign each triplet of digits as the ASCII codes for characters, pi somewhere contains this post, your post, in fact the entire forum, plus the DNA of everything living creature on the Earth, plus our entire history and every book ever written and every conversation ever engaged in. An infinite number of times.

We don't actually know if that's true. Just because it's infinite and non-repeating doesn't mean it has to contain every possible sequence of digits.

We don't actually know if that's true. Just because it's infinite and non-repeating doesn't mean it has to contain every possible sequence of digits.

Grant Hutchison

This kind of stuff makes my head hurt...

We can't know for sure either way, I get that, I think... But would it not be sensible to assume that it must? Since that its non repeating and is infinite then should we assume that every possible sequence of digits is in there? Or is it because the possible sequence of digits is infinite also which gives it the ambiguity?

We can't know for sure either way, I get that, I think... But would it not be sensible to assume that it must? Since that its non repeating and is infinite then should we assume that every possible sequence of digits is in there? Or is it because the possible sequence of digits is infinite also which gives it the ambiguity?

I think mathematicians believe it's likely that it does, but the last time I looked no-one had proved that it does. We don't even know if all digits 0 to 9 occur an infinite number of times in the decimal expansion of pi. It might just stop having nines at some point, for instance, leaving us with only a finite number of nines.
And it's trivially easy to come up with an infinite, non-repeating sequence that doesn't contain all finite sequences - 0110001111000001111110000000... for instance.

We can't know for sure either way, I get that, I think... But would it not be sensible to assume that it must? Since that its non repeating and is infinite then should we assume that every possible sequence of digits is in there?

It's easy to find counter examples. For instance, Liouville's constant is transcendental, irrational of course, infinite and non-repeating, but has only zeroes, and isolated single digits of 1, except for the first two digits.

Or is it because the possible sequence of digits is infinite also which gives it the ambiguity?

ETA:

Originally Posted by grant hutchison

I think mathematicians believe it's likely that it does, but the last time I looked no-one had proved that it does. We don't even know if all digits 0 to 9 occur an infinite number of times in the decimal expansion of pi. It might just stop having nines at some point, for instance, leaving us with only a finite number of nines.
And it's trivially easy to come up with an infinite, non-repeating sequence that doesn't contain all finite sequences - 0110001111000001111110000000... for instance.

Similarly, Liouville's constant is .1100010000000000100000... where the nth 1 is at the n! position.

In mathematics, this property is called normality. A normal number is a real number in which every sequence of digits occurs with the same frequency as every other possible sequence of digits with the same length. It's possible to prove that "almost all" (that term has a specific mathematical meaning in this case) real numbers are normal, but it turns out to be tantalizingly hard to show that any specific real number is or is not normal. As grant points out, although most mathematicians suspect that pi is normal, it has not been proven to be so.

In mathematics, this property is called normality. A normal number is a real number in which every sequence of digits occurs with the same frequency as every other possible sequence of digits with the same length. It's possible to prove that "almost all" (that term has a specific mathematical meaning in this case) real numbers are normal, but it turns out to be tantalizingly hard to show that any specific real number is or is not normal. As grant points out, although most mathematicians suspect that pi is normal, it has not been proven to be so.

The math involved in normal-numbers is way beyond my IQ, but with my naive interpretation, it should be possible to exist a not-normal number that contains any given finite sequence of digits, infinite times, just not with a uniform density. Same way a loaded Die will land 100 consecutive 1s, just much more rarely than it lands with 100 consecutive 6s.
There might be sequences which never occur in pi, but not being normal may or may not be relevant.
Corrections/insights are appreciated.